OFFSET
1,2
COMMENTS
This constant arises from a different interpretation of the equation x^^x = x+1, where x^^x indicates the tetration on the base x having the same height.
The alternative way to define x^^x is described by Takeji Ueda in his paper on Arxiv (see link below).
This definition implies that if Im(x) != 0, x cannot be a solution.
There are no other real solutions (conjecture).
LINKS
Takeji Ueda, Extension of tetration to real and complex heights, arXiv:2105.00247 [math.CA], 2021.
EXAMPLE
1.8441629749016...
MATHEMATICA
RealDigits[x /. FindRoot[x^(x^(((Log[x])^(x - 1) - 1)/(Log[x] - 1))) == x + 1, {x, 2}, WorkingPrecision -> 100]] [[1]]
PROG
(PARI) solve(x=3/2, 2, x^(x^(((log(x))^(x-1) - 1)/(log(x) - 1))) - x - 1) \\ Michel Marcus, Aug 29 2022
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Flavio Niccolò Baglioni and Marco Ripà, Aug 29 2022
STATUS
approved